Question

The horizontal distance between two poles is 15m. The angle of depression of the top of the first pole as seen from the top of the second pole is 300. If the height of the second pole is 24m. Find the height of the first pole.

Answer 1

Hint- In this question, we use the concept of application of trigonometry. We use trigonometric ratio $\tan \theta $ and we know $\tan \theta $ is the ratio of perpendicular to base of any right angle triangle.

Complete step-by-step solution -

Given that height of second pole AC=24meters, distance between two poles CD=15meters and angle of depression from top of second pole to top of first pole is 300.
Let height of first pole is x meters, DE=x
Now, In $\vartriangle ABE$
$ \Rightarrow \tan {30^0} = \dfrac{{AB}}{{BE}}$
We can see from the figure, AB= (24-x) and BE=CD=15m (opposite side of rectangle BCDE).
$
   \Rightarrow \tan {30^0} = \dfrac{{24 - x}}{{15}} \\
   \Rightarrow \dfrac{1}{{\sqrt 3 }} = \dfrac{{24 - x}}{{15}} \\
   \Rightarrow \dfrac{{15}}{{\sqrt 3 }} = 24 - x \\
   \Rightarrow x = 24 - \dfrac{{15}}{{\sqrt 3 }} \\
 $
We use rationalization,
$
   \Rightarrow x = 24 - \dfrac{{15}}{{\sqrt 3 }} \times \dfrac{{\sqrt 3 }}{{\sqrt 3 }} \\
   \Rightarrow x = 24 - \dfrac{{15\sqrt 3 }}{3} \\
   \Rightarrow x = 24 - 5\sqrt 3 \\
 $
Use value of $\sqrt 3 = 1.732$
$
   \Rightarrow x = 24 - 5 \times 1.732 \\
   \Rightarrow x = 24 - 8.66 \\
   \Rightarrow x = 15.34meters \\
$
So, the height of the first pole is 15.34meters.

Note- In such types of problems we use some important points to solve questions in an easy way. First draw the figure of all things mentioned in the question and mark the all sides, angle. Then apply $\tan \theta $ in a right angle triangle which gives the relation between heights of first pole and second pole. So, after calculation we will get the required answer.